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Full text of "Mathematical And Physical Papers - Iii"

11    ON THE  EFFECT OF  THE  INTERNAL FRICTION  OF FLUIDS
d       .        d      cos co d         Id,            A
-7- = sin co -7= — 1 --- 7- ğ = — r~ when co = U, a#               cfe       -or   a&>       OT aw
d2      1   d      1   d2     ,              n
— = — ~ — — - -j— when co =.Q, <Lz     ix dm    tx du>
whence we obtain
dp __    ./<Fu     d*u     1 du\       du ~     +       +~
(18).
Eliminating p from (16) and (17), and putting for /* its equivalent /jfp, we get
.
-or                  dx W     d^
JL-l(^-
dt \cZt3r    dx
or         i -, Q -]   -, - <) r — ~T   • aw
By virtue of (18), -or (iid-cr — qdx) is an exact differential.    Let then
-BT (^t!J — ^3?) = dty...................(20).
Expressing u and q in terms of ^, we get
cfe    ^     1 / d2      d2      1   cZ \
Substituting in (19); and operating separately on the factor -cr"1, we obtain
_L_           _a.         _             f-ft     C9n'^
.dğ2 "*" dw2    Ğr rfiB-    /*' At) \dtf + d^   -aM Y~   ' '^    >' Since the operations represented by the two expressions within parentheses are evidently convertible, the integral of this equation is
........................ (21),
* If we denote for shortness the operation
JL    JL. "'2